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On countably compact quasi-pseudometrizable spaces

Part of: Spaces with richer structures Fairly general properties

Published online by Cambridge University Press:  09 April 2009

Sergio Salbany  and
Salvador Romaguera
Sergio Salbany
Affiliation:
Department of MathematicsUniversity of ZimbabweP. O. Box MP 167 Mount Pleasant, Harare, Zimbabwe
Salvador Romaguera
Affiliation:
Department o de MatemáticaAplicadaEscuela de CaminosUniversity Politécnica46071 Valencia, Spain
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Abstract

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We prove the following results: (1) A quasi-metrizable space is compact if and only if every compatible quasi-metric has a quasi-metric left d—sequential completion. (2) A quasipseudometrizable space is countably compact if and only if every compatible quasi-pseudometric is pointwise bounded. (3) A quasi-pseudometrizable space is compact if and only if every compatible quasi-pseudometric is precompact.

Keywords

left d—Cauchy sequence, left d—(weakly, sequentially) complete quasipseudometric spacequasi-metric completion(countably) compact space

MSC classification

Secondary: 54D30: Compactness 54E35: Metric spaces, metrizability 54E45: Compact (locally compact) metric spaces 54E50: Complete metric spaces 54E52: Baire category, Baire spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 49 , Issue 2 , October 1990 , pp. 231 - 240
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Carlson, J. W. and Hicks, T. L., ‘On completences in a quasi-uniform space’, J. Math. Anal. Appl. 34 (1971), 618–627.CrossRefGoogle Scholar
[2]Ferrer, J. and Gregori, V., ‘A sequentially compact non-compact quasi-pseudometric space’, Monatsh. Math. 96 (1983), 269–270.CrossRefGoogle Scholar
[3]Fletcher, P. and Lindgren, W. F., Quasi-uniform spaces, (Marcel Dekker, New York, 1982).Google Scholar
[4]Kelley, J. L., General topology, (Springer-Verlag, New York, 1955).Google Scholar
[5]Künzi, H. P., ‘A note on sequentially compact quasi-pseudo-metric spaces’, Monatsh. Math. 95 (1983), 219–220.CrossRefGoogle Scholar
[6]Künzi, H. P., ‘Complete quasi-pseudo-metric spaces’, to appear.Google Scholar
[7]Reilly, I. L., Subrahmanyam, P. V. and Vamanamurthy, M. K., ‘Cauchy sequences in quasi-pseudo-metric spaces’, Monatsh. Math. 93 (1982), 127–140.CrossRefGoogle Scholar
[8]Romaguera, S. and Gutiérrez, A., ‘A note on Cauchy sequences in quasi-pseudo-metric spaces’, Glas. Mat. 21 (1986), 191–200.Google Scholar
[9]Romaguera, S. and Salbany, S., ‘On the Niemytzki-Tychonoff theorem for distance spaces’, to appear.Google Scholar